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    You are here: Modules > Iterations
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    Calculate algebraic series such as e = 1+ 1/2! + 1/3! + ..., a square wave, Fibonacci numbers. Study iterative maps, e.g. the (one-dimensional) Logistic map: (see below) or more complicated multi-dimensional maps. The logistic map is perhaps one of the simples mathematical system showing many characteristics of the development of chaotic behaviour.
    Several analytical tools are valailable to study the results of Iterations and ODE's such as:      
         time series | Power spectra | 2D projections | Fixed points | Lyapunov exponents
    A special window has been added in Mathgrapher v2 to allow detailed presentation of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.

    Logistic map: Sensitivity to initial conditions Projection in 2D Power spectrum Bifurcation diagram Lyapunov exponents
    Henon map: Definition 2D orbit Region of Stability
    Mandelbrot and Julia sets: Definition Mandelbrot: vary parameters Julia: vary initial conditions

    Projection in 2D

    applied to the logistic map:

    Choose a pair of F's from the Functions that are iterated and push Draw to make a
    graph where the X-axis represents the first and the Y-axis the second Function.

    The example below is a graph of F1 and F2 for a series of 1024 iterations of the logistic map.
    The first 100 iterations are omitted. Note that a 4-cycle has developed for a=0.87.

    Same as above for a=0.89 showing an 8-cycle and a=0.96 (the gap in the bifurcation diagram)
    where we have a 3-cycle. The behaviour is chaotic above 0.96.

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